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Chapter 8: Potential Energy and
Conservation of Energy
Work and kinetic energy are energies of motion.
rf
v v
1
1
2
2
∆K = K f − K i = mv f − mvi = Wnet = ∫ F ⋅ d r
2
2
ri
Consider a vertical spring oscillating with mass m
attached to one end. At the extreme ends of travel
the kinetic energy is zero, but something caused it
to accelerate back to the equilibrium point.
We need to introduce an energy that depends on
location or position. This energy is called potential
energy.
Chapter 8:
Potential Energy and Conservation
of Energy
• Work done by gravitation for a ball
thrown upward then fell down
Wab+ Wba = − mgd + mg d = 0
The gravitational force is said to be a
conservative force.
• A force is a conservative force if the net
work it does on a particle moving
around every closed path is zero.
b
a
Conservative forces
Wab,1 + Wba,2 = 0
Wab,2 + Wba,2 = 0
therefore: Wab,1 = Wab,2
i.e. The work done by a conservative force on a
particle moving between two points does not
depend on the path taken by the particle
So, ... choose the easiest path!!
Conservative & Non-conservative forces
Conservative Forces: (path independent)
gravitational
spring
electrostatic
Non-conservative forces: (dependent on path)
friction
air resistance
electrical resistance
These forces will convert mechanical energy into
heat and/or deformation.
Gravitation Potential Energy
• Potential energy is associated with the
configuration of a system in which a conservative
force acts:
∆U = −W
• For a general conservative force F = F(x)
xf v
∆U = − W = − ∫ F( x ) ⋅ dx̂
xi
• Gravitational potential energy:
∆U = − ∫ (−mg)dy = mg(y f − y i )
yf
yi
assume Ui = 0 at yi = 0 (reference point)
U(y) = mgy (gravitational potential energy)
only depends on vertical position
Nature only considers changes in Potential energy
important. Thus, we will always measure ∆U.
So, . . . We need to set a reference point!
The potential at that point could be defined to be
zero (i.e., Uref. point = 0) in which case we drop the
“∆” for convenience.
BUT, it is always understood to be there!
• Potential energy and reference point
A 0.5 kg physics book falls from a table 1.0 m to the ground.
What is U and ∆U if we take reference point y = 0 (assume
U = 0 at y = 0) at the ground?
y
∆U = Uf − Ui = −(mgh)
Physics
y=h
0
The book lost potential energy.
Actually, it was converted to
kinetic energy
h
y=0
• Potential energy and reference point
A 0.5 kg physics book falls from a table 1.0 m to the ground.
What is U and ∆U if we take reference point y = 0 (assume
U = 0 at y = 0) at the table?
y
∆U = Uf − Ui = mg(−h)
Physics
y=0
0
The book still the same lost
potential energy.
h
y = −h
• Sample problem 8-1: A block of cheese, m = 2 kg,
slides along frictionless track from a to b, the cheese
traveled a total distance of 2 m along the track, and a
net vertical distance of 0.8 m. How much is Wg?
b
Wnet
v v
= ∫ Fg ⋅ d r
a
But, ... We don’t know the
angle between F and dr
• Sample problem 8-1: A block of cheese, m = 2 kg,
slides along frictionless track from a to b, the cheese
traveled a total distance of 2 m along the track, and a
net vertical distance of 0.8 m. How much is Wg?
c
Easier (or another way)?
Split the problem into two
parts since gravity is
conservative
}=d
c
Wnet
b
v
v
= ∫ Fg ⋅ dx̂ + ∫ Fg ⋅ dŷ
a
0
c
mg(b-c) = mgd
Elastic Potential Energy
• Spring force is also a conservative force
F = −kx
xf
∆U = − W = ∫ (− kx )dx
xi
Uf – Ui = ½ kxf2 – ½ kxi2
Choose the free end of the relaxed spring as the
reference point:
that is: Ui = 0 at xi = 0
U = ½ kx2 Elastic potential energy
Conservation of Mechanical Energy
• Mechanical energy
Emec = K + U
• For an isolated system (no external forces), if there are
only conservative forces causing energy transfer within
the system….
We know: ∆K = W
(work-kinetic energy theorem)
Also:
∆U = −W
(definition of potential energy)
Therefore: ∆K + ∆U = 0 Î (Kf – Ki) + (Uf – Ui) = 0
therefore K1 + U1 = K2 + U2
Emec,1 = Emec,2 the mechanical energy is conserved
Daily Quiz Sept 22, 2004
• The figure shows four situations: one in which an
initially stationary block is dropped and three in
which the block is allowed to slide down
frictionless ramps. Which situation will the block
have the greatest kinetic energy at B?
5) all have the same kinetic energy at B
Daily Quiz Sept 22, 2004
Greatest Kinetic Energy
5) all have the same kinetic energy at B
Daily Quiz Sept 22, 2004
• The figure shows four situations: one in which an
initially stationary block is dropped and three in
which the block is allowed to slide down
frictionless ramps. Which situation will the block
have the greatest kinetic energy at B?
5) all have the same kinetic energy at B
The bowling ball pendulum demonstration
Mechanical
energy is
conserved if
there are only
conservative
forces acting
on the system.